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Kalman decomposition
Mathematical Theory From Wikipedia, the free encyclopedia
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In control theory, a Kalman decomposition provides a mathematical means to convert a representation of any linear time-invariant (LTI) control system to a form in which the system can be decomposed into a standard form which makes clear the observable and controllable components of the system. This decomposition results in the system being presented with a more illuminating structure, making it easier to draw conclusions on the system's reachable and observable subspaces.
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Definition
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Consider the continuous-time LTI control system
- ,
- ,
or the discrete-time LTI control system
- ,
- .
The Kalman decomposition is defined as the realization of this system obtained by transforming the original matrices as follows:
- ,
- ,
- ,
- ,
where is the coordinate transformation matrix defined as
- ,
and whose submatrices are
- : a matrix whose columns span the subspace of states which are both reachable and unobservable.
- : chosen so that the columns of are a basis for the reachable subspace.
- : chosen so that the columns of are a basis for the unobservable subspace.
- : chosen so that is invertible.
It can be observed that some of these matrices may have dimension zero. For example, if the system is both observable and controllable, then , making the other matrices zero dimension.
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Consequences
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By using results from controllability and observability, it can be shown that the transformed system has matrices in the following form:
This leads to the conclusion that
- The subsystem is both reachable and observable.
- The subsystem is reachable.
- The subsystem is observable.
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Variants
A Kalman decomposition also exists for linear dynamical quantum systems. Unlike classical dynamical systems, the coordinate transformation used in this variant requires to be in a specific class of transformations due to the physical laws of quantum mechanics.[1]
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